Κλασσικός Αρμονικός Ταλαντωτής
Κλασσικός Αρμονικός Ταλαντωτής Classical harmonic oscillator, Simple Harmonic Oscillator thumb|300px| [[Κλασσικός Αρμονικός Ταλαντωτής Κβαντικός Αρμονικός Ταλαντωτής ]] thumb|300px| [[Κλασσικός Αρμονικός Ταλαντωτής ]] - Ένας Αρμονικός Ταλαντωτής. Ετυμολογία To όνομα "Ταλαντωτής" προέρχεται ή συνδέεται ετυμολογικά με την λέξη "ταλάντωση". Εισαγωγή In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F'', proportional to the displacement, ''x: : \vec F = -k \vec x \, where k'' is a positive constant. If ''F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude). If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the friction coefficient, the system can: * Oscillate with a frequency smaller than in the non-damped case, and an amplitude decreasing with time (underdamped oscillator). * Decay to the equilibrium position, without oscillations (overdamped oscillator). The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called "critically damped." If an external time dependent force is present, the harmonic oscillator is described as a driven oscillator. Mechanical examples include pendula (with small angles of displacement), masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators such as RLC circuits. The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in Nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves. Ταξινομία Διακρίνουμε κυρίως τρία είδη αρμονικών ταλαντωτών: * Απλός Αρμονικός Ταλαντωτής (Simple) * Καθοδηγούμενος Αρμονικός Ταλαντωτής (εξαναγκαζόμενος) (Driven) * Αποσβεννύμενος Αρμονικός Ταλαντωτής (Dumped) Simple harmonic oscillator A simple harmonic oscillator is an oscillator that is neither driven nor damped. It consists of a mass m'', which experiences a single force, ''F, which pulls the mass in the direction of the point x''=0 and depends only on the mass's position ''x and a constant k''. Newton's second law for the system is : F = m a = m \frac{\mathrm{d}^2x}{\mathrm{d}t^2} = -k x. Solving this differential equation, we find that the motion is described by the function : x(t) = A\cos\left( 2\pi f t+\phi\right), where : f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} = \frac{1}{T}. The motion is periodic— repeating itself in a sinusoidal fashion with constant amplitude, ''A. In addition to its amplitude, the motion of a simple harmonic oscillator is characterized by its period T'', the time for a single oscillation or its frequency ''f = , the number of cycles per unit time. The position at a given time t'' also depends on the phase, ''φ, which determines the starting point on the sine wave. The period and frequency are determined by the size of the mass m'' and the force constant ''k, while the amplitude and phase are determined by the starting position and velocity. The velocity and acceleration of a simple harmonic oscillator oscillate with the same frequency as the position but with shifted phases. The velocity is maximum for zero displacement, while the acceleration is in the opposite direction as the displacement. The potential energy stored in a simple harmonic oscillator at position x'' is : U = \frac{1}{2}kx^2. Application to a conservative force The problem of the simple harmonic oscillator occurs frequently in physics, because a mass at equilibrium under the influence of any conservative force, in the limit of small motions, behaves as a simple harmonic oscillator. A conservative force is one that has a potential energy function. The potential energy function of a harmonic oscillator is: : V(x) = \frac{1}{2} k x^2 Given an arbitrary potential energy function V(x) , one can do a Taylor expansion in terms of x around an energy minimum ( x = x_0 ) to model the behavior of small perturbations from equilibrium. : V(x) = V(x_0) + (x-x_0) V'(x_0) + \frac{1}{2} (x-x_0)^2 V^{(2)}(x_0) + O(x-x_0)^3 Because V(x_0) is a minimum, the first derivative evaluated at x_0 must be zero, so the linear term drops out: : V(x) = V(x_0) + \frac{1}{2} (x-x_0)^2 V^{(2)}(x_0) + O(x-x_0)^3 The constant term ''V(x''0) is arbitrary and thus may be dropped, and a coordinate transformation allows the form of the simple harmonic oscillator to be retrieved: : V(x) \approx \frac{1}{2} x^2 V^{(2)}(0) = \frac{1}{2} k x^2 Thus, given an arbitrary potential energy function V(x) with a non-vanishing second derivative, one can use the solution to the simple harmonic oscillator to provide an approximate solution for small perturbations around the equilibrium point. Examples Simple pendulum exhibits simple harmonic motion under the conditions of no damping and small amplitude.]] Assuming no damping and small amplitudes, the differential equation governing a simple pendulum is : {\mathrm{d}^2\theta\over \mathrm{d}t^2}+{g\over \ell}\theta=0. The solution to this equation is given by: : \theta(t) = \theta_0\cos\left(\sqrt{g\over \ell}t\right) \quad\quad\quad\quad |\theta_0| \ll 1 where \theta_0 is the largest angle attained by the pendulum. The period, the time for one complete oscillation, is given by 2\pi divided by whatever is multiplying the time in the argument of the cosine ( \sqrt{g\over \ell} here). : T_0 = 2\pi\sqrt{\ell\over g}\quad\quad\quad\quad |\theta_0| \ll 1. Pendulum swinging over turntable Simple harmonic motion can in some cases be considered to be the one-dimensional projection of two-dimensional circular motion. Consider a long pendulum swinging over the turntable of a record player. On the edge of the turntable there is an object. If the object is viewed from the same level as the turntable, a projection of the motion of the object seems to be moving backwards and forwards on a straight line orthogonal to the view direction, sinusoidally like the pendulum. Spring/mass system When a spring is stretched or compressed by a mass, the spring develops a restoring force. Hooke's law gives the relationship of the force exerted by the spring when the spring is compressed or stretched a certain length: : F \left( t \right) =-kx \left( t \right) where ''F is the force, k'' is the spring constant, and ''x is the displacement of the mass with respect to the equilibrium position. This relationship shows that the distance of the spring is always opposite to the force of the spring. By using either force balance or an energy method, it can be readily shown that the motion of this system is given by the following differential equation: : F(t) = -kx(t) = m \frac {\mathrm{d}^{2}}{\mathrm{d}{t}^{2}} x \left( t \right) = ma. ...the latter evidently being Newton's second law of motion. If the initial displacement is A, and there is no initial velocity, the solution of this equation is given by: : x \left( t \right) =A \cos \left( \sqrt{k \over m}t \right). Given an ideal massless spring, m is the mass on the end of the spring. If the spring itself has mass, its effective mass must be included in m . Radial harmonic oscillator To solve for the orbit under a radial harmonic oscillator potential, it's easier to work in components r''' = (x, y, z). The potential energy can be written : V(\mathbf{r}) = \frac{1}{2} kr^{2} = \frac{1}{2} k \left( x^{2} + y^{2} + z^{2}\right) The equation of motion for a particle of mass m'' is given by three independent Lagrange's equations : \frac{d^{2}x}{dt^{2}} + \omega_{0}^{2} x = 0 : \frac{d^{2}y}{dt^{2}} + \omega_{0}^{2} y = 0 : \frac{d^{2}z}{dt^{2}} + \omega_{0}^{2} z = 0 where the constant \omega_{0}^{2} \equiv \frac{k}{m} must be positive (i.e., ''k > 0) to ensure bounded, closed orbits; otherwise, the particle will fly off to infinity. The solutions of these simple harmonic oscillator equations are all similar : x = A_{x} \cos \left(\omega_{0} t + \phi_{x} \right) : y = A_{y} \cos \left(\omega_{0} t + \phi_{y} \right) : z = A_{z} \cos \left(\omega_{0} t + \phi_{z} \right) where the positive constants Ax, Ay and Az represent the '''amplitudes of the oscillations and the angles φ''x'', φ''y'' and φ''z'' represent their phases. The resulting orbit r(t'') = ''y(y), z(t) is closed because it repeats exactly after a period : T \equiv \frac{2\pi}{\omega_{0}} The system is also stable because small perturbations in the amplitudes and phases cause correspondingly small changes in the overall orbit. Υποσημειώσεις Εσωτερική Αρθρογραφία *Ταλαντωτής *Ταλάντωση *Κβαντικός Αρμονικός Ταλαντωτής *Απλή Αρμονική Ταλάντωση Βιβλιογραφία Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *[ ] *[ ] Category: Κυματική Φυσική